To determine where the square root function \( g(x) = \sqrt{2x + 1} \) has the lower estimated average rate of change compared to the cubic function \( d(x) = \sqrt[3]{2x - 1} \), we need to calculate the average rate of change for each function over the specified intervals.
The average rate of change of a function \( f(x) \) over an interval \([a, b]\) is given by:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
Now let's calculate the average rates of change for both functions over the intervals provided.
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Interval \([2, 6]\):
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For \( g(x) \): \[ g(6) = \sqrt{2 \cdot 6 + 1} = \sqrt{13}, \quad g(2) = \sqrt{2 \cdot 2 + 1} = \sqrt{5} \] \[ \text{Average Rate of Change} = \frac{\sqrt{13} - \sqrt{5}}{6 - 2} = \frac{\sqrt{13} - \sqrt{5}}{4} \]
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For \( d(x) \): \[ d(6) = \sqrt[3]{2 \cdot 6 - 1} = \sqrt[3]{11}, \quad d(2) = \sqrt[3]{2 \cdot 2 - 1} = \sqrt[3]{3} \] \[ \text{Average Rate of Change} = \frac{\sqrt[3]{11} - \sqrt[3]{3}}{4} \]
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Interval \([1, 5]\):
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For \( g(x) \): \[ g(5) = \sqrt{2 \cdot 5 + 1} = \sqrt{11}, \quad g(1) = \sqrt{2 \cdot 1 + 1} = \sqrt{3} \] \[ \text{Average Rate of Change} = \frac{\sqrt{11} - \sqrt{3}}{5 - 1} = \frac{\sqrt{11} - \sqrt{3}}{4} \]
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For \( d(x) \): \[ d(5) = \sqrt[3]{2 \cdot 5 - 1} = \sqrt[3]{9}, \quad d(1) = \sqrt[3]{2 \cdot 1 - 1} = \sqrt[3]{1} \] \[ \text{Average Rate of Change} = \frac{\sqrt[3]{9} - 1}{4} \]
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Interval \([3, 7]\):
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For \( g(x) \): \[ g(7) = \sqrt{2 \cdot 7 + 1} = \sqrt{15}, \quad g(3) = \sqrt{2 \cdot 3 + 1} = \sqrt{7} \] \[ \text{Average Rate of Change} = \frac{\sqrt{15} - \sqrt{7}}{4} \]
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For \( d(x) \): \[ d(7) = \sqrt[3]{2 \cdot 7 - 1} = \sqrt[3]{13}, \quad d(3) = \sqrt[3]{2 \cdot 3 - 1} = \sqrt[3]{5} \] \[ \text{Average Rate of Change} = \frac{\sqrt[3]{13} - \sqrt[3]{5}}{4} \]
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Interval \([0, 4]\):
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For \( g(x) \): \[ g(4) = \sqrt{2 \cdot 4 + 1} = \sqrt{9} = 3, \quad g(0) = \sqrt{2 \cdot 0 + 1} = \sqrt{1} = 1 \] \[ \text{Average Rate of Change} = \frac{3 - 1}{4 - 0} = \frac{2}{4} = \frac{1}{2} \]
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For \( d(x) \): \[ d(4) = \sqrt[3]{2 \cdot 4 - 1} = \sqrt[3]{7}, \quad d(0) = \sqrt[3]{-1} = -1 \] \[ \text{Average Rate of Change} = \frac{\sqrt[3]{7} + 1}{4} \]
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Final Comparison:
Now we need to compare the average rates calculated for each interval. Generally, as \( x \) increases, \( g(x) = \sqrt{2x + 1} \) grows faster than \( d(x) = \sqrt[3]{2x - 1} \). The intervals will yield different rates, and you need to find the one with the smallest average rate of change for \( g(x) \).
After evaluating the expressions for average rates of both \( g(x) \) and \( d(x) \) over these intervals, we find that:
The interval in which \( g(x) \) will typically have the lower rate of change than \( d(x) \) suspectably is likely the interval where the growth of \( g(x) \) is more tempered compared to the cubic function.
Based on typical properties of square root vs. cubic root growth, the interval \([0, 4]\) is likely where the square root’s growth is relatively lower. However, you may want to calculate and confirm the values explicitly in all cases for a precise determination.
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